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Passband modulation
Analog modulation
AM FM PM QAM SM SSB
Digital modulation
ASK APSK CPM FSK MFSK MSK OOK PPM PSK QAM SC-FDE TCM TC-PAM WDM
Hierarchical modulation
QAM WDM
Spread spectrum
CSS DSSS FHSS THSS
See also
Capacity-approaching codes Demodulation Line coding Modem AnM PoM PAM PCM PDM PWM ΔΣM OFDM FDM Multiplexing
v t e

Non-orthogonal frequency-division multiplexing (N-OFDM) is a method of encoding digital data on multiple carrier frequencies with non-orthogonal intervals between frequency of sub-carriers.^[1][2][3] N-OFDM signals can be used in communication and radar systems.

The low-pass equivalent N-OFDM signal is expressed as:^[3][2]

${\displaystyle \nu (t)=\sum _{k=0}^{N-1}X_{k}e^{j2\pi \alpha kt/T},\quad 0\leq t<T,}$

where ${\displaystyle X_{k}}$ are the data symbols, ${\displaystyle N}$ is the number of sub-carriers, and ${\displaystyle T}$ is the N-OFDM symbol time. The sub-carrier spacing ${\displaystyle \alpha /T}$ for ${\displaystyle \alpha <1}$ makes them non-orthogonal over each symbol period.

The history of N-OFDM signals theory was started in 1992 from the Patent of Russian Federation No. 2054684.^[1] In this patent, Vadym Slyusar proposed the 1st method of optimal processing for N-OFDM signals after Fast Fourier transform (FFT).

In this regard need to say that W. Kozek and A. F. Molisch wrote in 1998 about N-OFDM signals with ${\displaystyle \alpha <1}$ that "it is not possible to recover the information from the received signal, even in the case of an ideal channel."^[4]

In 2001, V. Slyusar proposed non-orthogonal frequency digital modulation (N-OFDM) as an alternative of OFDM for communications systems.^[5]

The next publication about this method has priority in July 2002^[2] before the conference paper regarding SEFDM of I. Darwazeh and M.R.D. Rodrigues (September, 2003).^[6]

Despite the increased complexity of demodulating N-OFDM signals compared to OFDM, the transition to non-orthogonal subcarrier frequency arrangement provides several advantages:

1. higher spectral efficiency, which allows to reduce the frequency band occupied by the signal and improve the electromagnetic compatibility of many terminals;
2. adaptive detuning from interference concentrated in frequency by changing the nominal frequencies of the subcarriers;^[7]
3. an ability to take into account Doppler frequency shifts of subcarriers when working with subscribers moving at high speeds;
4. reduction of the peak factor of the multi-frequency signal mixture.

This section describes a simple idealized N-OFDM system model suitable for a time-invariant AWGN channel.^[8]

An N-OFDM carrier signal is the sum of a number of not-orthogonal subcarriers, with baseband data on each subcarrier being independently modulated commonly using some type of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier.

${\displaystyle s[n]}$ is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into ${\displaystyle \scriptstyle N}$ parallel streams, and each one mapped to a (possibly complex) symbol stream using some modulation constellation (QAM, PSK, etc.). Note that the constellations may be different, so some streams may carry a higher bit-rate than others.

A Digital Signal Processor (DSP) is computed on each set of symbols, giving a set of complex time-domain samples. These samples are then quadrature-mixed to passband in the standard way. The real and imaginary components are first converted to the analogue domain using digital-to-analogue converters (DACs); the analogue signals are then used to modulate cosine and sine waves at the carrier frequency, ${\displaystyle f_{\text{c}}}$, respectively. These signals are then summed to give the transmission signal, ${\displaystyle s(t)}$.

The receiver picks up the signal ${\displaystyle r(t)}$, which is then quadrature-mixed down to baseband using cosine and sine waves at the carrier frequency. This also creates signals centered on ${\displaystyle 2f_{\text{c}}}$, so low-pass filters are used to reject these. The baseband signals are then sampled and digitised using analog-to-digital converters (ADCs), and a forward FFT is used to convert back to the frequency domain.

This returns ${\displaystyle N}$ parallel streams, which use in appropriate symbol detector.

The 1st method of optimal processing for N-OFDM signals after FFT was proposed in 1992.^[1]

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The method of optimal processing for N-OFDM signals without FFT was proposed in October 2003.^[3][9] In this case can be used ADC samples.

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The combination N-OFDM and MIMO technology is similar to OFDM. To the building of MIMO system can be used digital antenna array as transmitter and receiver of N-OFDM signals.

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Fast-OFDM^[10][11][12] method was proposed in 2002.^[13]

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Filter-bank multi-carrier modulation (FBMC) is.^[14][15][16] As example of FBMC can consider Wavelet N-OFDM.

N-OFDM has become a technique for power-line communications (PLC). In this area of research, a wavelet transform is introduced to replace the DFT as the method of creating non-orthogonal frequencies. This is due to the advantages wavelets offer, which are particularly useful on noisy power lines.^[17]

To create the sender signal the wavelet N-OFDM uses a synthesis bank consisting of a ${\displaystyle N}$-band transmultiplexer followed by the transform function

${\displaystyle F_{n}(z)=\sum _{k=0}^{L-1}f_{n}(k)z^{-k},\quad 0\leq n<N}$

On the receiver side, an analysis bank is used to demodulate the signal again. This bank contains an inverse transform

${\displaystyle G_{n}(z)=\sum _{k=0}^{L-1}g_{n}(k)z^{-k},\quad 0\leq n<N}$

followed by another ${\displaystyle N}$-band transmultiplexer. The relationship between both transform functions is

${\displaystyle {\begin{aligned}f_{n}(k)&=g_{n}(L-1-k)\\F_{n}(z)&=z^{-(L-1)}G_{n}*(z-1)\end{aligned}}}$

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N-OFDM is a spectrally efficient method.^[6][18] All SEFDM methods are similar to N-OFDM.^{[6][19][20][21][22][23][24]}

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Generalized frequency division multiplexing (GFDM) is.

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• OFDM
• COFDM